Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B

Average Accuracy: 91.4% → 94.6%

Time: 11.8s

Precision: binary64

Cost: 7304

\[ \begin{array}{c}[z, t] = \mathsf{sort}([z, t])\\ \end{array} \]

Math

\[\frac{x}{y - z \cdot t} \]

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+162}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-1}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ x (- y (* z t))))

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -1e+162)
   (* (/ x t) (/ -1.0 z))
   (if (<= (* z t) 2e+277) (/ x (fma (- z) t y)) (/ (- (/ x t)) z))))

C

double code(double x, double y, double z, double t) {
	return x / (y - (z * t));
}

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -1e+162) {
		tmp = (x / t) * (-1.0 / z);
	} else if ((z * t) <= 2e+277) {
		tmp = x / fma(-z, t, y);
	} else {
		tmp = -(x / t) / z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	return Float64(x / Float64(y - Float64(z * t)))
end

↓

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -1e+162)
		tmp = Float64(Float64(x / t) * Float64(-1.0 / z));
	elseif (Float64(z * t) <= 2e+277)
		tmp = Float64(x / fma(Float64(-z), t, y));
	else
		tmp = Float64(Float64(-Float64(x / t)) / z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -1e+162], N[(N[(x / t), $MachinePrecision] * N[(-1.0 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+277], N[(x / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], N[((-N[(x / t), $MachinePrecision]) / z), $MachinePrecision]]]

Target

Original	91.4%
Target	93.1%
Herbie	94.6%

\[\begin{array}{l} \mathbf{if}\;x < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{elif}\;x < 2.1378306434876444 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -9.9999999999999994e161

   1. Initial program 86.2%

      \[\frac{x}{y - z \cdot t} \]

   2. Applied egg-rr 85.8%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{x}{y - z \cdot t}}\right)}^{3}} \]
      Step-by-step derivation

      [Start] 86.2	\[ \frac{x}{y - z \cdot t} \]
      add-cube-cbrt [=>] 85.7	\[ \color{blue}{\left(\sqrt[3]{\frac{x}{y - z \cdot t}} \cdot \sqrt[3]{\frac{x}{y - z \cdot t}}\right) \cdot \sqrt[3]{\frac{x}{y - z \cdot t}}} \]
      pow3 [=>] 85.8	\[ \color{blue}{{\left(\sqrt[3]{\frac{x}{y - z \cdot t}}\right)}^{3}} \]

   3. Taylor expanded in y around 0 86.1%

      \[\leadsto \color{blue}{-1 \cdot \left({1}^{0.3333333333333333} \cdot \frac{x}{t \cdot z}\right)} \]

   4. Simplified 99.7%

      \[\leadsto \color{blue}{-\frac{\frac{x}{t}}{z}} \]
      Step-by-step derivation

      [Start] 86.1	\[ -1 \cdot \left({1}^{0.3333333333333333} \cdot \frac{x}{t \cdot z}\right) \]
      mul-1-neg [=>] 86.1	\[ \color{blue}{-{1}^{0.3333333333333333} \cdot \frac{x}{t \cdot z}} \]
      pow-base-1 [=>] 86.1	\[ -\color{blue}{1} \cdot \frac{x}{t \cdot z} \]
      *-lft-identity [=>] 86.1	\[ -\color{blue}{\frac{x}{t \cdot z}} \]
      associate-/r* [=>] 99.7	\[ -\color{blue}{\frac{\frac{x}{t}}{z}} \]

   5. Applied egg-rr 99.8%

      \[\leadsto -\color{blue}{\frac{x}{t} \cdot \frac{1}{z}} \]
      Step-by-step derivation

      [Start] 99.7	\[ -\frac{\frac{x}{t}}{z} \]
      div-inv [=>] 99.8	\[ -\color{blue}{\frac{x}{t} \cdot \frac{1}{z}} \]

   if -9.9999999999999994e161 < (*.f64 z t) < 2.00000000000000001e277

   1. Initial program 94.7%

      \[\frac{x}{y - z \cdot t} \]

   2. Applied egg-rr 94.8%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(-z, t, y\right)}} \]
      Step-by-step derivation

      [Start] 94.7	\[ \frac{x}{y - z \cdot t} \]
      sub-neg [=>] 94.7	\[ \frac{x}{\color{blue}{y + \left(-z \cdot t\right)}} \]
      +-commutative [=>] 94.7	\[ \frac{x}{\color{blue}{\left(-z \cdot t\right) + y}} \]
      distribute-lft-neg-in [=>] 94.7	\[ \frac{x}{\color{blue}{\left(-z\right) \cdot t} + y} \]
      fma-def [=>] 94.8	\[ \frac{x}{\color{blue}{\mathsf{fma}\left(-z, t, y\right)}} \]

   if 2.00000000000000001e277 < (*.f64 z t)

   1. Initial program 68.9%

      \[\frac{x}{y - z \cdot t} \]

   2. Applied egg-rr 68.8%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{x}{y - z \cdot t}}\right)}^{3}} \]
      Step-by-step derivation

      [Start] 68.9	\[ \frac{x}{y - z \cdot t} \]
      add-cube-cbrt [=>] 68.8	\[ \color{blue}{\left(\sqrt[3]{\frac{x}{y - z \cdot t}} \cdot \sqrt[3]{\frac{x}{y - z \cdot t}}\right) \cdot \sqrt[3]{\frac{x}{y - z \cdot t}}} \]
      pow3 [=>] 68.8	\[ \color{blue}{{\left(\sqrt[3]{\frac{x}{y - z \cdot t}}\right)}^{3}} \]

   3. Taylor expanded in y around 0 68.9%

      \[\leadsto \color{blue}{-1 \cdot \left({1}^{0.3333333333333333} \cdot \frac{x}{t \cdot z}\right)} \]

   4. Simplified 99.9%

      \[\leadsto \color{blue}{-\frac{\frac{x}{t}}{z}} \]
      Step-by-step derivation

      [Start] 68.9	\[ -1 \cdot \left({1}^{0.3333333333333333} \cdot \frac{x}{t \cdot z}\right) \]
      mul-1-neg [=>] 68.9	\[ \color{blue}{-{1}^{0.3333333333333333} \cdot \frac{x}{t \cdot z}} \]
      pow-base-1 [=>] 68.9	\[ -\color{blue}{1} \cdot \frac{x}{t \cdot z} \]
      *-lft-identity [=>] 68.9	\[ -\color{blue}{\frac{x}{t \cdot z}} \]
      associate-/r* [=>] 99.9	\[ -\color{blue}{\frac{\frac{x}{t}}{z}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+162}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-1}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	95.4%
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -4 \cdot 10^{+302}:\\ \;\;\;\;\frac{-1}{t} \cdot \frac{x}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternative 2
Accuracy	94.6%
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+162}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-1}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \end{array} \]

Alternative 3
Accuracy	67.4%
Cost	913

\[\begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+73}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -1800000 \lor \neg \left(y \leq -1 \cdot 10^{-37}\right) \land y \leq 82000000:\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4
Accuracy	68.8%
Cost	913

\[\begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+73}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -3.2 \lor \neg \left(y \leq -1.1 \cdot 10^{-23}\right) \land y \leq 85000000:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5
Accuracy	69.0%
Cost	912

\[\begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{+73}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq -950:\\ \;\;\;\;-\frac{\frac{x}{z}}{t}\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-23}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 26500000:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6
Accuracy	52.3%
Cost	452

\[\begin{array}{l} \mathbf{if}\;t \leq 4.8 \cdot 10^{+225}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{t}\\ \end{array} \]

Alternative 7
Accuracy	49.6%
Cost	320

\[\frac{1}{\frac{y}{x}} \]

Alternative 8
Accuracy	50.0%
Cost	192

\[\frac{x}{y} \]

Reproduce

herbie shell --seed 2023157 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< x -1.618195973607049e+50) (/ 1.0 (- (/ y x) (* (/ z x) t))) (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) (/ 1.0 (- (/ y x) (* (/ z x) t)))))

  (/ x (- y (* z t))))